Subgroups — C₂×#16811232 #20288080

Order 6784.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811232	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133216	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133212	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133212	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133212	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133211	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133208	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133207	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133207	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133207	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13133195	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133195	C₂	1	1	yes	maximal, characteristic
13	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#13133182	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#13133175	C₂	1	1	yes	maximal, characteristic
16	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
20	#13133216	C₂²	1	8	yes	complemented
21	#13133212	C₂²	1	8	yes	—
22	#13133212	C₂²	1	8	yes	—
23	#13133212	C₂²	1	8	yes	—
24	#13133211	C₂²	1	8	yes	—
25	#13133208	C₂²	1	8	yes	complemented
26	#13133207	C₂²	1	8	yes	complemented
27	#13133207	C₂²	1	8	yes	complemented
28	#13133207	C₂²	1	8	yes	complemented
29	#13133195	C₂²	1	8	yes	—
30	#13133195	C₂²	1	8	yes	—
31	C₄×Dic₁₀₆	C₂²	1	8	yes	complemented
32	#13133182	C₂²	1	8	yes	complemented
33	#13133175	C₂²	1	8	yes	—
34	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
38	C₂×#11719919	C₂²	1	1	yes	characteristic
39	C₂×#11719919	C₂²	1	1	yes	characteristic
40	C₂×#11719919	C₂²	1	1	yes	characteristic
41	C₂×#11719919	C₂²	1	1	yes	characteristic
42	C₂×#11719919	C₂²	1	1	yes	characteristic
43	C₂×#11719919	C₂²	1	1	yes	characteristic
44	C₂×#11719919	C₂²	1	1	yes	characteristic
45	C₂×#11719919	C₂²	1	1	yes	characteristic
46	C₂×#11719919	C₂²	1	1	yes	characteristic
47	C₂×#11719919	C₂²	1	1	yes	characteristic
48	C₂×#11719921	C₂²	1	1	yes	characteristic
49	C₂×#11719921	C₂²	1	1	yes	characteristic
50	C₂×#11719921	C₂²	1	1	yes	characteristic
51	C₂×#11719921	C₂²	1	1	yes	characteristic
52	C₂×#11719921	C₂²	1	1	yes	characteristic
53	C₂×#11719921	C₂²	1	1	yes	characteristic
54	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
55	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
56	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
57	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
58	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
59	#125×C₅₃	C₂²	1	8	yes	complemented
60	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
67	C₂³×D₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
76	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
77	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
78	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
80	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
81	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₄×D₅₃	—	2	4	no	—
83	C₂×C₄×D₅₃	C₂³	1	4	yes	—
84	C₂×Dic₁₀₆	—	2	4	no	—
85	C₂×Dic₁₀₆	—	2	4	no	—
86	C₂×Dic₁₀₆	C₂³	1	4	yes	—
87	C₂×Dic₁₀₆	C₂³	1	4	yes	—
88	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
90	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
91	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
92	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
93	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₄²×C₅₃	C₂³	1	4	yes	—
95	#11719921	C₂³	1	4	yes	—
96	#11719921	C₂³	1	4	yes	—
97	#11719921	C₂³	1	4	yes	—
98	#11719921	C₂³	1	4	yes	—
99	#11719921	C₂³	1	4	yes	—
100	#11719921	C₂³	1	4	yes	—
101	C₄⋊Dic₅₃	C₂³	1	4	yes	—
102	C₄⋊Dic₅₃	C₂³	1	4	yes	—
103	C₄⋊Dic₅₃	C₂³	1	4	yes	—
104	C₄⋊Dic₅₃	C₂³	1	4	yes	—
105	#11719919	C₂³	1	4	yes	—
106	#11719919	C₂³	1	4	yes	—
107	#11719919	C₂³	1	4	yes	—
108	#11719919	C₂³	1	4	yes	—
109	#11719919	C₂³	1	4	yes	—
110	#11719919	C₂³	1	4	yes	—
111	#11719919	C₂³	1	4	yes	—
112	#11719919	C₂³	1	4	yes	—
113	#11719919	C₂³	1	4	yes	—
114	#11719919	C₂³	1	4	yes	—
115	C₄×Dic₅₃	C₂³	1	4	yes	—
116	C₄×Dic₅₃	C₂³	1	4	yes	—
117	C₄×Dic₅₃	C₂³	1	4	yes	—
118	C₄×Dic₅₃	C₂³	1	4	yes	—
119	C₄×Dic₅₃	C₂³	1	4	yes	—
120	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
121	C₂²×D₅₃	—	2	2	no	—
122	C₂²×D₅₃	—	2	2	no	—
123	C₂²×D₅₃	—	2	2	no	—
124	C₂²×D₅₃	C₂⁴	1	2	yes	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
133	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
134	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
135	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
136	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
137	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
138	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
139	C₂×Dic₅₃	—	2	2	no	—
140	C₂×Dic₅₃	—	2	2	no	—
141	C₂×Dic₅₃	—	2	2	no	—
142	C₂×Dic₅₃	—	2	2	no	—
143	C₂×Dic₅₃	—	2	2	no	—
144	C₂×Dic₅₃	—	2	2	no	—
145	C₂×Dic₅₃	#41	1	4	yes	—
146	C₂×Dic₅₃	C₂⁴	1	2	yes	—
147	C₂×Dic₅₃	C₂⁴	1	2	yes	—
148	C₂×Dic₅₃	C₂⁴	1	2	yes	—
149	C₂×Dic₅₃	C₂⁴	1	2	yes	—
150	C₂×Dic₅₃	C₂⁴	1	2	yes	—
151	C₂×Dic₅₃	C₂⁴	1	2	yes	—
152	C₂×Dic₅₃	C₂⁴	1	2	yes	—
153	C₄×D₅₃	—	2	8	no	—
154	Dic₁₀₆	—	2	8	no	—
155	Dic₁₀₆	—	2	8	no	—
156	C₂²×C₅₃	C₂×#41	1	2	yes	—
157	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
158	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
159	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
160	C₂×D₅₃	—	2	2	no	—
161	C₂×D₅₃	—	2	2	no	—
162	C₂×D₅₃	—	2	2	no	—
163	C₂×D₅₃	—	4	4	no	—
164	C₄×C₅₃	—	2	2	no	—
165	C₄×C₅₃	—	2	2	no	—
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	2	2	no	—
171	Dic₅₃	—	2	2	no	—
172	Dic₅₃	—	2	2	no	—
173	Dic₅₃	—	2	2	no	—
174	Dic₅₃	—	2	2	no	—
175	Dic₅₃	—	2	2	no	—
176	Dic₅₃	—	2	2	no	—
177	Dic₅₃	C₂×#41	1	4	yes	—
178	C₂×#541	—	53	1	no	maximal, Sylow
179	C₂×C₅₃	#541	1	4	yes	complemented
180	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
181	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
182	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
183	D₅₃	—	4	2	no	—
184	#541	—	53	16	no	—
185	C₂×C₄⋊Q₈	—	53	1	no	—
186	C₂×#126	—	53	1	no	—
187	C₂×#126	—	53	1	no	—
188	C₂×#126	—	53	1	no	—
189	C₂×#126	—	53	1	no	—
190	C₂×#125	—	53	1	no	—
191	C₂×#125	—	53	1	no	—
192	C₂×#125	—	53	1	no	—
193	C₂×#125	—	53	1	no	—
194	C₂×#125	D₅₃	1	1	yes	characteristic, complemented
195	C₂×C₂²⋊Q₈	—	53	1	no	—
196	C₂×C₂²⋊Q₈	—	53	1	no	—
197	C₂×C₄×Q₈	—	53	1	no	—
198	C₂×C₄×Q₈	—	53	1	no	—
199	C₂×#117	—	53	1	no	—
200	C₅₃	C₂×#541	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	—	53	1	no	—
202	C₂²×Q₈	—	53	1	no	—
203	C₂³×C₄	—	53	1	no	—
204	C₄⋊Q₈	—	53	8	no	—
205	#126	—	53	8	no	—
206	#126	—	53	8	no	—
207	#126	—	53	8	no	—
208	#126	—	53	8	no	—
209	#125	—	53	8	no	—
210	#125	—	53	8	no	—
211	#125	—	53	8	no	—
212	#125	—	53	8	no	—
213	#125	C₂×D₅₃	1	8	yes	complemented
214	C₂²⋊Q₈	—	53	8	no	—
215	C₂²⋊Q₈	—	53	8	no	—
216	C₄×Q₈	—	53	8	no	—
217	C₄×Q₈	—	53	8	no	—
218	#117	—	53	8	no	—
219	C₂×C₄⋊C₄	—	53	1	no	—
220	C₂×C₄⋊C₄	—	53	1	no	—
221	C₂×C₄⋊C₄	—	53	1	no	—
222	C₂×C₄⋊C₄	—	53	1	no	—
223	C₂×C₄⋊C₄	—	53	1	no	—
224	C₂×C₄⋊C₄	—	53	1	no	—
225	C₂×C₄⋊C₄	—	53	1	no	—
226	C₂×C₄⋊C₄	—	53	1	no	—
227	C₂×C₄⋊C₄	—	53	1	no	—
228	C₂×C₄⋊C₄	—	53	1	no	—
229	C₂×C₄⋊C₄	—	53	1	no	—
230	C₂×C₄⋊C₄	—	53	1	no	—
231	C₂×C₄⋊C₄	—	53	1	no	—
232	C₂×C₄⋊C₄	—	53	1	no	—
233	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
235	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
236	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
237	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
238	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	—	53	1	no	—
240	C₂×C₂²⋊C₄	—	53	1	no	—
241	C₂×C₂²⋊C₄	—	53	1	no	—
242	C₂×C₂²⋊C₄	—	53	1	no	—
243	C₂×C₂²⋊C₄	—	53	1	no	—
244	C₂×C₂²⋊C₄	—	53	1	no	—
245	C₂×C₄²	—	53	1	no	—
246	C₂×C₄²	—	53	1	no	—
247	C₂×C₄²	—	53	1	no	—
248	C₂×C₄²	—	53	1	no	—
249	C₂×C₄²	—	53	1	no	—
250	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
251	C₂⁴	—	53	1	no	—
252	C₂×Q₈	—	53	4	no	—
253	C₂×Q₈	—	53	4	no	—
254	C₂×Q₈	—	106	4	no	—
255	C₂×Q₈	—	106	4	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	53	1	no	—
258	C₂²×C₄	—	53	1	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	1	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	53	1	no	—
263	C₂²×C₄	—	53	4	no	—
264	C₂²×C₄	—	106	4	no	—
265	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
272	C₄⋊C₄	—	53	4	no	—
273	C₄⋊C₄	—	53	4	no	—
274	C₄⋊C₄	—	53	4	no	—
275	C₄⋊C₄	—	53	4	no	—
276	C₄⋊C₄	—	53	4	no	—
277	C₄⋊C₄	—	53	4	no	—
278	C₄⋊C₄	—	53	4	no	—
279	C₄⋊C₄	—	53	4	no	—
280	C₄⋊C₄	—	53	4	no	—
281	C₄⋊C₄	—	53	4	no	—
282	C₄⋊C₄	—	53	4	no	—
283	C₄⋊C₄	—	53	4	no	—
284	C₄⋊C₄	—	53	4	no	—
285	C₄⋊C₄	—	53	4	no	—
286	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
287	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
288	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
289	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
290	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
291	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
292	C₂²⋊C₄	—	53	4	no	—
293	C₂²⋊C₄	—	53	4	no	—
294	C₂²⋊C₄	—	53	4	no	—
295	C₂²⋊C₄	—	53	4	no	—
296	C₂²⋊C₄	—	53	4	no	—
297	C₂²⋊C₄	—	53	4	no	—
298	C₄²	—	53	4	no	—
299	C₄²	—	53	4	no	—
300	C₄²	—	53	4	no	—
301	C₄²	—	53	4	no	—
302	C₄²	—	53	4	no	—
303	C₄²	C₂²×D₅₃	1	4	yes	—
304	C₂³	—	53	2	no	—
305	C₂³	—	106	2	no	—
306	C₂³	—	106	2	no	—
307	C₂³	—	106	2	no	—
308	C₂³	C₂³×D₅₃	1	1	yes	characteristic
309	Q₈	—	106	8	no	—
310	Q₈	—	106	8	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	53	4	no	—
319	C₂×C₄	—	53	2	no	—
320	C₂×C₄	—	53	2	no	—
321	C₂×C₄	—	53	2	no	—
322	C₂×C₄	—	53	2	no	—
323	C₂×C₄	—	53	2	no	—
324	C₂×C₄	—	53	2	no	—
325	C₂×C₄	—	53	2	no	—
326	C₂×C₄	—	106	8	no	—
327	C₂×C₄	—	106	2	no	—
328	C₂×C₄	—	106	2	no	—
329	C₂×C₄	—	106	2	no	—
330	C₂×C₄	—	106	2	no	—
331	C₂×C₄	—	106	2	no	—
332	C₂×C₄	—	106	2	no	—
333	C₂×C₄	C₂³×D₅₃	1	2	yes	—
334	C₂×C₄	C₂³×D₅₃	1	2	yes	—
335	C₂×C₄	C₂³×D₅₃	1	2	yes	—
336	C₂×C₄	C₂³×D₅₃	1	2	yes	—
337	C₂×C₄	C₂³×D₅₃	1	2	yes	—
338	C₂×C₄	C₂³×D₅₃	1	2	yes	—
339	C₂×C₄	C₂³×D₅₃	1	2	yes	—
340	C₂²	—	106	2	no	—
341	C₂²	—	106	2	no	—
342	C₂²	—	106	2	no	—
343	C₂²	—	212	4	no	—
344	C₂²	#41×D₅₃	1	2	yes	—
345	C₂²	#13133264	1	2	yes	—
346	C₂²	#13133265	1	2	yes	—
347	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	53	4	no	—
356	C₄	—	106	2	no	—
357	C₄	—	106	2	no	—
358	C₄	—	106	2	no	—
359	C₄	—	106	2	no	—
360	C₄	—	106	2	no	—
361	C₄	—	106	2	no	—
362	C₂	—	212	2	no	—
363	C₂	#16811232	1	4	yes	direct product factor
364	C₂	C₂×#41×D₅₃	1	1	yes	characteristic
365	C₂	C₂×#13133264	1	1	yes	characteristic
366	C₂	C₂×#13133265	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×#125×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	C₂×#541
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #16811232	C₂	direct (N × H)
2. C₂×#13133216	C₂	semidirect
7. C₂×#13133208	C₂	semidirect
8. C₂×#13133207	C₂	semidirect
9. C₂×#13133207	C₂	semidirect
10. C₂×#13133207	C₂	semidirect
13. C₂×C₄×Dic₁₀₆	C₂	semidirect
14. C₂×#13133182	C₂	semidirect
16. C₂×#125×C₅₃	C₂	semidirect
20. #13133216	C₂²	semidirect
25. #13133208	C₂²	semidirect
26. #13133207	C₂²	semidirect
27. #13133207	C₂²	semidirect
28. #13133207	C₂²	semidirect
31. C₄×Dic₁₀₆	C₂²	semidirect
32. #13133182	C₂²	semidirect
59. #125×C₅₃	C₂²	semidirect
179. C₂×C₅₃	#541	semidirect
194. C₂×#125	D₅₃	semidirect
200. C₅₃	C₂×#541	semidirect
213. #125	C₂×D₅₃	semidirect
363. C₂	#16811232	direct (N × H)

Subgroups — C2×#16811232 #20288080

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811232 #20288080